Subordinators.

(Subordination) Let
be a Markov process,
and
is a stochastic process
.
In addition, we assume that
and
are independent. We define the "subordinated" process
.

The process
does not need to be Markovian.

We proceed to investigate conditions for
to be Markovian.

Let
be the transition function for the process
and
be the transition function for the process
.
We calculate the transition function for the subordinated process
.
For
we define a uniform mesh over
with step
and apply the formula
(
Total_probability_rule
):
By independence of
and
we
simplify
We need the expression
Prob
to be well defined for any
and any however fine mesh
.
This means that the law of
needs to be independent of
.
Thus
needs to be a function of the form
We
continue
Next, we would like to
write
This imposes regularity requirements
along
-parameter.
We pass to the limit
:
For
to be Markovian it needs to satisfy the formula
(
Kolmogorov-Chapman equation
),
:
We calculate the
integral
Note that by assumption,
is Markovian and has the formula
(
Kolmogorov-Chapman
equation
):
Hence, we
continue
We make a change of variables
and change the order of
integration
For
to be Markovian the last expression should be equal to
thus we
need
The convolution on the right is a formula for distribution of a sum of two
independent r.v. We conclude that the increments of
should be infinitely divisible and stationary. We summarize these findings in
the following proposition.

Proposition

Let
be a Markov process with
-differentiable
transition function
and
is a Levy process with non-negative increments. Then the subordinated process
is Markovian.